LINEAR

See rule-classes for a general discussion of rule classes and how they are
used to build rules from formulas. An example :corollary formula
from which a :linear rule might be built is:

Example:
(implies (and (eqlablep e) if inequality reasoning begins to
(true-listp x)) consider how (length (member a b))
(<= (length (member e x)) compares to any other term, add to
(length x))) set of known inequalities the fact
that it is no larger than (length b),
provided (eqlablep a) and (true-listp b)
rewrite to t

Note: One :linear rule class object might create many linear
arithmetic rules from the :corollary formula. To create the rules,
we first flatten the and and implies structure of the formula,
transforming it into a conjunction of formulas, each of the form

(implies (and h1 ... hn) (rel lhs rhs))

where no hypothesis is a conjunction and rel is one of the
inequality relations <, <=, =, >, or >=. If
necessary, the hypothesis of such a conjunct may be vacuous.
We create a :linear rule for each such conjunct, if possible, and
otherwise cause an error.

Each rule has one or more ``trigger
terms'' which may be specified by the user using the :trigger-terms
field of the rule class or which may be defaulted to values chosen
by the system. We discuss the determination of trigger terms after
discussing how linear rules are used.

Recall that we eliminate the unknowns of an inequality in
term-order, largest unknowns first. (See term-order.) In order to
facilitate this strategy, we store the inequalities in ``linear pots''.
For purposes of the present discussion, let us say that an inequality is
``about'' its largest unknown. Then, all of the inequalities about
a particular unknown are stored in the same linear pot, and the pot
is said to be ``labeled'' with that unknown. This storage layout
groups all of the inequalities which are potential candidates for
cancellation with each other into one place. It is also key to the
efficient operation of :linear rules.

If the arithmetic decision procedure has stabilized and not yielded a
contradiction, we scan through the list of linear pots examining
each label as we go. If the trigger term of some :linear rule
can be instantiated to match the label, we so instantiate that rule
and attempt to relieve the hypotheses with general-purpose rewriting.
If we are successful, we add the rule's instantiated conclusion to
our set of inequalities. This may let cancellation continue.

Note: Problems may arise if you explicitly store a linear lemma
under a trigger term that, when instantiated, is not the largest
unknown in the instantiated concluding inequality. Suppose for
example you store the linear rule (<= (fn i j) (/ i (* j j))) under
the trigger term (fn i j). Then when the system ``needs'' an
inequality about (fn a b), (i.e., because (fn a b) is the
label of some linear pot, and hence the largest unknown in some
inequality), it
will appeal to the rule and deduce (<= (fn a b) (/ a (* b b))).
However, the largest unknown in this inequality is (/ a (* b b)) and
hence it will be stored in a linear pot labeled with (/ a (* b b)).
The original, triggering inequality which is in a pot about (fn a b)
will therefore not be cancelled against the new one. It is generally
best to specify as a trigger term one of the ``maximal'' terms of the
polynomial, as described below.

We now describe how the trigger terms are determined. Most of the
time, the trigger terms are not specified by the user and are
instead selected by the system. However, the user may specify the
terms by including an explicit :trigger-terms field in the rule
class, e.g.,

Each termi must be a term and must not be a variable, quoted
constant, lambda application, let-expression or if-expression. In
addition, each termi must be such that if all the variables in the
term are instantiated and then the hypotheses of the corollary
formula are relieved (possibly instantiating additional free
variables), then all the variables in the concluding inequality are
instantiated. We generate a linear rule for each conjuctive branch
through the corollary and store each rule under each of the
specified triggers. Thus, if the corollary formula contains several
conjuncts, the variable restrictions on the termi must hold for each
conjunct.

If :trigger-terms is omitted the system computes a set of trigger
terms. Each conjunct of the corollary formula may be given a unique
set of triggers depending on the variables that occur in the
conjunct and the addends that occur in the concluding inequality.
In particular, the trigger terms for a conjunct is the list of all
``maximal addends'' in the concluding inequality.

The ``addends'' of (+ x y) and (- x y) are the union of the
addends of x and y. The addends of (- x) and (* n x),
where n is a rational constant, is just {x}. The
addends of an inequality are the union of the addends of the left-
and right-hand sides. The addends of any other term, x, is
{x}.

A term is maximal for a conjunct (implies hyps concl) of the
corollary if (a) the term is a non-variable, non-quote, non-lambda
application, non-let and non-if expression, (b) the term
contains enough variables so that when they are instantiated and the
hypotheses are relieved (which may bind some free variables;
see free-variables) then all the variables in concl are
instantiated, and (c) no other addend is always ``bigger'' than the
term, in the technical sense described below.

The technical notion below depends on the notion of fn-count,
the number of function symbols in a term, and pseudo-fn-count,
which is the number of function symbols implicit in a constant (see
the comment on pseduo-fn-count in the definition of var-fn-count in
the sources for details). We say term1 is always bigger than
term2 if all instances of term1 have a larger fn-count
(actually lexicographic order of fn-count and pseudo-fn-count) than
the corresponding instances of term2. This is equivalent to
saying that the fn-count of term1 is larger than that of
term2 (by ``fn-count'' here we mean the lexicographic order of
fn-count and pseudo-fn-count) and the variable bag for term2 is
a subbag of that for term1. For example, (/ a (* b b)) is
always bigger than (fn a b) because the first has two function
applications and {a b} is a subbag of a b b, but
(/ a (* b b)) is not always bigger than (fn a x).